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Ch. 2 - Descriptive Statistics
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Larson 8th EditionCh. 2 - Descriptive Statistics문제 2.4.22c
2장, 문제 2.4.22c

Graphical Analysis In Exercises 21–24, you are asked to compare three data sets.


(c) Estimate the sample standard deviations. Then determine how close each of your estimates is by finding the sample standard deviations.


i.Bar graph showing frequency distribution of data entries ranging from 4 to 10, with peaks at 6 and 7.
ii. Bar graph showing frequency distribution of data entries from 4 to 10, with peaks at 4, 5, 9, and 10.
iii. Bar graph showing frequency distribution with data entries ranging from 4 to 10, highlighting values around 6 to 8.

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Step 1: Observe the three histograms provided. Each histogram represents a different data set. The spread and concentration of the data entries will help estimate the sample standard deviation. Wider spreads typically indicate higher standard deviations, while narrower spreads suggest lower standard deviations.
Step 2: For the first histogram (i), the data entries are spread across the range 4 to 10, with a peak frequency at 7. The distribution appears moderately spread out, suggesting a medium standard deviation. To calculate the sample standard deviation, use the formula: \( s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \), where \( x_i \) are the data points, \( \bar{x} \) is the mean, and \( n \) is the sample size.
Step 3: For the second histogram (ii), the data entries are concentrated at the extremes (4 and 10) with minimal frequencies in the middle. This indicates a larger spread and likely a higher standard deviation compared to the first histogram. Apply the same formula for standard deviation, ensuring to account for the extreme values.
Step 4: For the third histogram (iii), the data entries are tightly concentrated around 6, 7, and 8, with no entries outside this range. This suggests a very narrow spread and a low standard deviation. Use the standard deviation formula, noting the limited range of values.
Step 5: After estimating the standard deviations for each histogram, calculate the exact values using the formula provided. Compare the calculated values to your initial estimates to determine how close your estimates were to the actual standard deviations.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Sample Standard Deviation

The sample standard deviation is a measure of the amount of variation or dispersion in a set of sample data points. It quantifies how much the individual data points deviate from the sample mean. A low standard deviation indicates that the data points tend to be close to the mean, while a high standard deviation indicates a wider spread of values.
추천 영상:
08:45
Calculating Standard Deviation

Frequency Distribution

A frequency distribution is a summary of how often each value occurs in a dataset. It is often represented graphically using histograms, where the x-axis represents the data entries and the y-axis represents the frequency of those entries. This visual representation helps in understanding the distribution and central tendencies of the data.
추천 영상:
06:38
Intro to Frequency Distributions

Estimation Techniques

Estimation techniques involve using sample data to infer characteristics about a larger population. In the context of standard deviation, one might estimate the sample standard deviation based on visual inspection of the data distribution before calculating it precisely. This approach helps in making quick assessments and comparisons between different datasets.
추천 영상:
06:33
Introduction to Confidence Intervals
관련 실천
교과서 질문

Shifting Data Sample annual salaries (in thousands of dollars) for employees at a company are listed.

40   35   49   53   38   39   40

37   49   34   38   43   47   35


c. Each employee in the sample takes a pay cut of \$2000 from their original salary. Find the sample mean and the sample standard deviation for the revised data set.

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교과서 질문

Use the relative frequency histogram to describe any patterns with the data.

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교과서 질문

Use the data set and the indicated number of classes to construct


(c) a frequency polygon,

Pulse Rates

Number of classes: 6 Data set: Pulse rates of all students in a class 68 105 95 80 90 100 75 70 84 98 102 70 65 88 90 75 78 94 110 120 95 80 76 108

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교과서 질문

Life Spans of Tires A brand of automobile tire has a mean life span of 35,000 miles, with a standard deviation of 2250 miles. Assume the life spans of the tires have a bell-shaped distribution.


b. The life spans of three randomly selected tires are 30,500 miles, 37,250 miles, and 35,000 miles. Using the Empirical Rule, find the percentile that corresponds to each life span.

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교과서 질문

Extending Concepts


Trimmed Mean To find the 10% trimmed mean of a data set, order the data, delete the lowest 10% of the entries and the highest 10% of the entries, and find the mean of the remaining entries.


c. What is the benefit of using a trimmed mean versus using a mean found using all data entries? Explain your reasoning.

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교과서 질문

Extending Concepts


Golf The distances (in yards) for nine holes of a golf course are listed.

336 393 408 522 147 504 177 375 360


c. Compare the measures you found in part (b) with those found in part (a). What do you notice?

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